In this paper, we prove a collection of results on graphical indices. We determine the extremal graphs attaining the maximal generalized Wiener index (e.g. the hyper-Wiener index) among all graphs with given matching number or independence number. This generalizes some work of Dankelmann, as well as some work of Chung. We also show alternative proofs for two recents results on maximizing the Wiener index and external Wiener index by deriving it from earlier results. We end with proving two conjectures. We prove that the maximum for the difference of the Wiener index and the eccentricity is attained by the path if the order $n$ is at least $9$ and that the maximum weighted Szeged index of graphs of given order is attained by the balanced complete bipartite graphs.

• 05C09 - Graphical indices (Wiener index, Zagreb index, Randi? index, etc.)
• 05C12 - Distance in graphs
• 05C35 - Extremal problems in graph theory
• 05C92 - Chemical graph theory
• 92E10 - Molecular structure (graph-theoretic methods, methods of differential topology, etc.)